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The simple-root singular vector in a Verma module

Statement

Let $α_{i}$ be simple and put $m = ⟨ λ + ρ, α_{i}^{\lor} ⟩$ . If $m \in Z_{> 0}$ , then $f_{i}^{m} v_{λ}$ is a singular vector in $M (λ)$ of weight $s_{i} \cdot λ$ .

Facts & Assumptions

Given: The Verma convention Verma modules and its supplied triangular root-space decomposition, together with the reflection and Weyl-vector conventions Root reflections and the Weyl group action and The Weyl vector rho for a chosen positive system.

[L1]

The PBW model identifies $M (λ)$ with $U (n^{-}) v_{λ}$ as a vector space (The PBW model of a Verma module).

[L2]

For a maximal-toral subalgebra of a finite-dimensional complex semisimple Lie algebra, Finite semisimple Cartan, root and string structure supplies normalized root vectors $e_{i}, f_{i}$ with $[e_{i}, f_{i}] = α_{i}^{\lor}$ and the corresponding ${s l}_{2}$ relations.

Proof

technique · direct

1.1L1L2givenalgebra

The supplied root-space direct sum has zero weight space $h$ . Thus every $ad (h)$ is diagonalizable, $h$ is abelian, and its centralizer is its zero weight space $h$ ; hence it is maximal toral. Apply [L2] to choose nonzero $e_{i} \in g_{α_{i}}$ and $f_{i} \in g_{- α_{i}}$ with $[e_{i}, f_{i}] = α_{i}^{\lor}$ . The resulting ${s l}_{2}$ relations give, by induction, $e_{i} f_{i}^{r} v_{λ} = r (⟨ λ, α_{i}^{\lor} ⟩ - r + 1) f_{i}^{r - 1} v_{λ} .$ At $r = m = ⟨ λ, α_{i}^{\lor} ⟩ + 1$ this is zero, while [L1] shows $f_{i}^{m} v_{λ} \neq 0$ .

2.1step 1.1algebra∎

For $j \neq i$ , $[e_{j}, f_{i}] = 0$ because $α_{j} - α_{i}$ is not a root; hence every $e_{j}$ also kills $f_{i}^{m} v_{λ}$ . Its weight is $λ - m α_{i} = s_{i} (λ + ρ) - ρ$ , so it is singular of the stated dot weight.

Depends on

Used by

Equal central character does not give every Verma embedding direction Counterexample
Ordinary Weyl orbits do not classify central characters Counterexample
Dot-conjugate type-A₂ weights have the same central character Example
The BGG resolution for sl2 Example
The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma
Simple-reflection embeddings of Verma modules Proposition
Harish-Chandra isomorphism for the center Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, Representations of Lie Groups, Exercise 8.15(i) (standard reference, not scraped)